Existence of Positive Solutions to Nonlinear Fractional Boundary Value Problem with Changing Sign Nonlinearity and Advanced Arguments

Abstract

We discuss the existence of positive solutions to a class of fractional boundary value problem with changing sign nonlinearity and advanced arguments $x(\mathrm{0})={\mathrm{x}}^{\mathrm{\prime }}(\mathrm{0})=\mathrm{0},$ $x(\mathrm{1})=\beta x(\eta )+\lambda [x],\mathrm{}\mathrm{}\mathrm{}\mathrm{}\beta >\mathrm{0},\; and\mathrm{}\mathrm{}\mathrm{}\mathrm{}\eta \in (\mathrm{0,1}),$ where $D^{\alpha }$ is the standard Riemann-Liouville derivative, $f:[\mathrm{0},\mathrm{\infty })\to [\mathrm{0},\mathrm{\infty })$ is continuous, $f(\mathrm{0})>\mathrm{0}$,$\mathrm{}\mathrm{}\mathrm{\hspace{0.17em}h\hspace{0.17em}}:[\mathrm{0,1}]\to (-\mathrm{\infty },+\mathrm{\infty })$, and $\mathrm{}\mathrm{}a(t)$ is the advanced argument. Our analysis relies on a nonlinear alternative of Leray-Schauder type. An example is given to illustrate our results.